RIESZ TYPE KERNELS OVER THE RING OF INTEGERS OF A LOCAL-FIELD

Let O denote the ring of integers of a local field. In this note we pr ove an approximation theorem for the Riesz type kernels {gamma(mu,lamb da,n)}(n=1)(x) over O. The proof requires a sharp estimate of the Diri chlet kernel D-n(x) on P-1\O, which may also have independent interest . As a consequence we solve the local field analog of N. J. Fine's pro blem (Trans. Amer. Math. Sec. 65, 1949, 372-414) concerning rates of c onvergence of the Fejer sum on the Walsh system. Our approach has pote ntial applications to other operators such as Cesaro means and Abel-Po isson means of ''type I.'' (C) 1997 Academic Press.